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Properties

Label 21.3.44553216873...7527.2
Degree $21$
Signature $[3, 9]$
Discriminant $-\,3^{48}\cdot 7^{21}$
Root discriminant $86.23$
Ramified primes $3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T42

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2732211, 1545075, -1282302, -1587250, -874503, 982989, 943152, 658251, 97020, -58527, -87570, -30807, -11858, 0, -1089, -189, -252, -63, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 63*x^17 - 252*x^16 - 189*x^15 - 1089*x^14 - 11858*x^12 - 30807*x^11 - 87570*x^10 - 58527*x^9 + 97020*x^8 + 658251*x^7 + 943152*x^6 + 982989*x^5 - 874503*x^4 - 1587250*x^3 - 1282302*x^2 + 1545075*x + 2732211)
 
gp: K = bnfinit(x^21 - 63*x^17 - 252*x^16 - 189*x^15 - 1089*x^14 - 11858*x^12 - 30807*x^11 - 87570*x^10 - 58527*x^9 + 97020*x^8 + 658251*x^7 + 943152*x^6 + 982989*x^5 - 874503*x^4 - 1587250*x^3 - 1282302*x^2 + 1545075*x + 2732211, 1)
 

Normalized defining polynomial

\( x^{21} - 63 x^{17} - 252 x^{16} - 189 x^{15} - 1089 x^{14} - 11858 x^{12} - 30807 x^{11} - 87570 x^{10} - 58527 x^{9} + 97020 x^{8} + 658251 x^{7} + 943152 x^{6} + 982989 x^{5} - 874503 x^{4} - 1587250 x^{3} - 1282302 x^{2} + 1545075 x + 2732211 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-44553216873221843443058843338471726567527=-\,3^{48}\cdot 7^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.23$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{110226440709177665692630459556041699978154580027652757599476288506577} a^{20} - \frac{2192819233618575181622043200896577685054061685177185659380308163470}{36742146903059221897543486518680566659384860009217585866492096168859} a^{19} - \frac{3092086452912425764618637983903388329222189388314947823770508169929}{12247382301019740632514495506226855553128286669739195288830698722953} a^{18} - \frac{3127740072192531748059482027807974592418940886454026278321308980485}{12247382301019740632514495506226855553128286669739195288830698722953} a^{17} - \frac{4930673984184351836401490705099290426164897283308878853988734947313}{12247382301019740632514495506226855553128286669739195288830698722953} a^{16} - \frac{3924620377241770869341123442240447597131244566152157572251305294483}{12247382301019740632514495506226855553128286669739195288830698722953} a^{15} - \frac{4348125802948209776509585973174401927029467926682698043197614397727}{12247382301019740632514495506226855553128286669739195288830698722953} a^{14} + \frac{377519879060814255492806287994233307144530960341897122738264470631}{12247382301019740632514495506226855553128286669739195288830698722953} a^{13} - \frac{1432286061448067028689417723995156344914774291397873783452735346177}{12247382301019740632514495506226855553128286669739195288830698722953} a^{12} + \frac{49012942143538691456459746373809329708642809391020206708085799351106}{110226440709177665692630459556041699978154580027652757599476288506577} a^{11} - \frac{8386259088797973194347144618376217784069133955279023307647424163783}{36742146903059221897543486518680566659384860009217585866492096168859} a^{10} + \frac{2585998446656767641749545590951057908320992370373347256168926525115}{12247382301019740632514495506226855553128286669739195288830698722953} a^{9} - \frac{6060447465324705697457727958814406318792547961172870025717331093020}{12247382301019740632514495506226855553128286669739195288830698722953} a^{8} - \frac{4914667650170488601462469830761819126218324493397223670852856498218}{12247382301019740632514495506226855553128286669739195288830698722953} a^{7} - \frac{4297033543084695475375134958817461218712018956197423275340542232496}{12247382301019740632514495506226855553128286669739195288830698722953} a^{6} + \frac{16781591621388004758126919807419703839063053432578575047307691187345}{36742146903059221897543486518680566659384860009217585866492096168859} a^{5} - \frac{4617143726068177436792405850554087536513119935848782849086724089245}{12247382301019740632514495506226855553128286669739195288830698722953} a^{4} - \frac{1704744348588195047560524255586089445999052730264920803136885688110}{12247382301019740632514495506226855553128286669739195288830698722953} a^{3} - \frac{31145429290161642095619343395081803838700810627727256066812286136690}{110226440709177665692630459556041699978154580027652757599476288506577} a^{2} + \frac{1407438949863783984295506173001699233996915442427600400897880586702}{36742146903059221897543486518680566659384860009217585866492096168859} a + \frac{1175751505589521943042986055834063616849699231620586869044078555050}{12247382301019740632514495506226855553128286669739195288830698722953}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1324635290840 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T42:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6174
The 37 conjugacy class representatives for t21n42
Character table for t21n42 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 21 siblings: data not computed
Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R $18{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ $18{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $18{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ $18{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
7Data not computed